Elastic waves in a glasslike disordered chain

Abstract

The attenuation of long-wavelength elastic waves is studied in a glasslike harmonic chain. The structural disorder gives rise to a phase incoherence of an initial plane-wave disturbance. In addition the dynamic disorder introduces a phase incoherence and an exponential localization of the normal modes. The phase-incoherence contribution can interfere with the structural disorder since the spring constant is strongly correlated with equilibrium separation. To calculate the attenuation the long-wavelength normal modes must be explicitly constructed. This is done by a slight modification of the state-ratio method of Matsuda and Ishii, and a straightforward extension of this method from the random mass chain to the random spring-constant chain. In the absence of structural disorder, the attenuation of the average displacement is shown to have equal contributions from the phase incoherence and from the localization of the modes. With structural disorder, when the spring constant is a decreasing function of equilibrium separation, the attenuation is decreased by the interference between dynamical and structural phase incoherence. The low-frequency limit of the density of states is also calculated. It is related to the sound speed exactly as in the ordered case.